A274634 -id:A274634 - cp's OEIS frontend

A284664 Number of proper colorings of the 2n-gon with 2 instances of each of n colors under rotational symmetry.

0, 1, 5, 96, 3528, 199620, 15908400, 1697149440, 233631921600, 40335000693120, 8536048230528000, 2173422135804796800, 655519296840629760000, 231135191421131129932800, 94208725354330431747302400, 43956400457238853734678528000, 23278422600113660887881093120000

Offset: 1

Marko Riedel, Mar 31 2017

nonn

			When n=2 the coloring of the nodes of the square with two instances each of black and white must alternate and a rotation by Pi/4 takes one coloring to the other, so there is just one coloring.

Omar Sehlouli, Marko Riedel, Hexagon coloring
Marko Riedel, Image of the five colorings of the hexagon (n=3).

Cf. A274634, A284747.

For n>=2, (1/2)(n-1)! + (1/(2n)) * Sum_{p=0..n} C(n,p) ((-1)^p/2^(n-p)) ((2n-p)! + p(2n-p-1)!).

A284747 Number of proper colorings of the 2n-gon with 2 instances of each of n colors under dihedral (rotational and reflectional) symmetry.

Original entry on oeis.org

0, 1, 4, 54, 1794, 99990, 7955460, 848584800, 116816051520, 20167501253760, 4268024125243200, 1086711068022148800, 327759648421871635200, 115567595710587359539200, 47104362677165542792243200, 21978200228619432098036736000, 11639211300056830532862403584000, 6943663015969522875618267601920000

Offset: 1

Marko Riedel, Apr 01 2017

nonn

			When n=2 the coloring of the nodes of the square with two instances each of black and white must alternate and a rotation by Pi/4 takes one coloring to the other, so there is just one coloring.

Omar Sehlouli, Marko Riedel, Hexagon coloring

Cf. A274634, A284664.

For n>=2, (1/4)(n-1)! + (1/4)n! + (1/(4n)) * Sum_{p=0..n} C(n,p) ((-1)^p/2^(n-p)) ((2n-p)! + p(2n-p-1)!).

cp's OEIS Frontend

A284664 Number of proper colorings of the 2n-gon with 2 instances of each of n colors under rotational symmetry.

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